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or
y _x b     a
Squaring the last equation and simplifying we have
cos 5 = sin2 5,
Y
9 a2
, fc2
(3)
~,A
which is the equation of an ellipse, Fig. 136. The following cases are of special interest.
Case I.   When 5 = 0, equation (3) reduces to y = - Xj which is the equation of a
the line AA\ Substituting the values of x and y in the equation
r = Vx2 + y2
we obtain         ______
r = Va2 + 62 • sin ut
for the equation of the motion".    Therefore the motion is simple harmonic, in the line AA', with an amplitude equal
,27T CO
FIG. 136.
to Va2 + fe2 and period'
Case II.   When 5 = x, equation (3) reduces to y= — -x.
a
Therefore the motion is similar to that in Case I and takes place in the line BBf.
Case III.   When 5 = ± £ equation (3) reduces to
2
~~o ~l~ r~9 = ^> <z      o
while equations (1) and (2) become
# = a sin co£, y — b cos out.
In this case, therefore, the particle describes an elliptical